Max-min fairness (MMF) is a widely known approach to a fair allocation of bandwidth to each of the users in a network. This allocation can be computed by uniformly raising the bandwidths of all users without violating capacity constraints. We consider an extension of these allocations by raising the bandwidth with arbitrary and not necessarily uniform time-depending velocities (allocation rates). These allocations are used in a game-theoretic context for routing choices, which we formalize in progressive filling games (PFGs). We present a variety of results for equilibria in PFGs. We show that these games possess pure Nash and strong equilibria. While computation in general is NP-hard, there are polynomial-time algorithms for prominent classes of Max-Min-Fair Games (MMFG), including the case when all users have the same source-destination pair. We characterize prices of anarchy and stability for pure Nash and strong equilibria in PFGs and MMFGs when players have different or the same source-destination pairs. In addition, we show that when a designer can adjust allocation rates, it is possible to design games with optimal strong equilibria. Some initial results on polynomial-time algorithms in this direction are also derived.
We initiate the study of dynamic traffic assignment for electrical vehicles addressing the specific challenges such as range limitations and the possibility of battery recharge at predefined charging locations. We pose the dynamic equilibrium problem within the deterministic queueing model of Vickrey and as our main result, we establish the existence of an energy-feasible dynamic equilibrium. There are three key modeling-ingredients for obtaining this existence result:1. We introduce a walk-based definition of dynamic traffic flows which allows for cyclic routing behavior as a result of recharging events en route.2. We use abstract convex feasibility sets in an appropriate function space to model the energy-feasibility of used walks.3. We introduce the concept of capacitated dynamic equilibrium walk-flows which generalize the former unrestricted dynamic equilibrium path-flows.Viewed in this framework, we show the existence of an energy-feasible dynamic equilibrium by applying an infinite dimensional variational inequality, which in turn requires a careful analysis of continuity properties of the network loading as a result of injecting flow into walks.We complement our theoretical results by a computational study in which we design a fixed-point algorithm computing energy-feasible dynamic equilibria. We apply the algorithm to standard real-world instances from the traffic assignment community illustrating the complex interplay of resulting travel times, energy consumption and prices paid at equilibrium.
The famous Braess paradox describes the following phenomenon: It might happen that the improvement of resources, like building a new street within a congested network, may in fact lead to larger costs for the players in an equilibrium. In this paper we consider general nonatomic congestion games and give a characterization of the maximal combinatorial property of strategy spaces for which Braess paradox does not occur. In a nutshell, bases of matroids are exactly this maximal structure. We prove our characterization by two novel sensitivity results for convex separable optimization problems over polymatroid base polyhedra which may be of independent interest.
In cost sharing games with delays, a set of agents jointly allocates a finite subset of resources. Each resource has a fixed cost that has to be shared by the players, and each agent has a nonshareable player-specific delay for each resource. A prominent example is uncapacitated facility location (UFL), where facilities need to be opened (at a shareable cost) and clients want to connect to opened facilities. Each client pays a cost share and his non-shareable physical connection cost. Given any profile of subsets allocated by the agents, a separable cost sharing protocol determines cost shares that satisfy budget balance on every resource and separability over the resources. Moreover, a separable protocol guarantees existence of pure Nash equilibria in the induced strategic game for the agents. In this paper, we study separable cost sharing protocols in several general combinatorial domains. We provide black-box reductions to reduce the design of a separable cost-sharing protocol to the design of an approximation algorithm for the underlying cost minimization problem. In this way, we obtain new separable cost-sharing protocols in games based on arbitrary player-specific matroids, single-source connection games without delays, and connection games on $n$-series-parallel graphs with delays. All these reductions are efficiently computable - given an initial allocation profile, we obtain a cheaper profile and separable cost shares turning the profile into a pure Nash equilibrium. Hence, in these domains any approximation algorithm can be used to obtain a separable cost sharing protocol with a price of stability bounded by the approximation factor.
Resource assignment problems occur in a vast variety of applications, from scheduling problems over image recognition to communication networks. Often these problems can be modeled by a maximum weight matching problem in (bipartite) graphs or generalizations thereof, and efficient and practical algorithms are known for these problems. Although in some of the applications an assignment of the resources may be needed only once, in many of these applications, the assignment has to be computed more often for different scenarios. In that case it is often essential that the assignments can be computed very fast. Moreover, implementing different assignments in different scenarios may come with a certain cost for the reconfiguration of the system. In this paper, we consider the problem of determining optimal assignments sequentially over a given time horizon, where consecutive assignments are coupled by constraints that control the cost of reconfiguration. We develop fast approximation and online algorithms for this problem with provable approximation guarantees and competitive ratios. Moreover, we present an extensive computational study about the applicability of our model and our algorithms in the context of orthogonal frequency division multiple access (OFDMA) wireless networks, finding a significant performance improvement for the total bandwidth of the system using our algorithms. For this application (the downlink of an OFDMA wireless cell) , the run time of matching algorithms is extremely important, having an acceptable range of a few milliseconds only. For the considered realistic instances, our algorithms perform extremely well: the solution quality is, on average, within a factor of 0.8–0.9 of optimal off-line solutions, and the running times are at most 5 ms per phase even in the worst case. Thus, our algorithms are well suited to be applied in the context of OFDMA systems. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2015.2221 . This paper was accepted by Teck-Hua Ho, optimization.
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